Related papers: Diametral diameter two properties in Banach spaces
We construct a Banach space $X$ with the r-BSP such that the infimum of the diameter of the slices of the unit ball is $1$, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by…
We introduce the super alternative Daugavet property (super ADP) which lies strictly between the Daugavet property and the Alternative Daugavet property as follows. A Banach space $X$ has the super ADP if for every element $x$ in the unit…
The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The…
This work explores the equivalence of two sequential properties, $D$ and $D'$, for dual Banach spaces under the weak* topology. Property $D$ ensures that any totally scalarly measurable function is also scalarly measurable, while property…
This is the first of two closely related papers on transversality. Here we introduce the notion of tangential transversality of two closed subsets of a Banach space. It is an intermediate property between transversality and…
Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the…
We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a…
Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analysed. A geometric description is given in terms of nonlinear slices. Some local versions of…
In this paper structure of infinite dimensional Banach spaces is studied by using an asymptotic approach based on stabilization at infinity of finite dimensional subspaces which appear everywhere far away. This leads to notions of…
In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where…
In this paper we study properties of complete sets and of completions of sets in Banach spaces. We consider the family of completions of a given set and its size; we also study in detail the relationships concerning diameters, radii, and…
We introduce a new diametral notion for points of the unit sphere of Banach spaces, that naturally complements the notion of Delta-points, but is weaker than the notion of Daugavet points. We prove that this notion can be used to provide a…
We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.
Necessary and sufficient conditions for Banach space to be(isometrically isomorphic to) a dual space will be given.
Property($M$) in separable Banach spaces has played an important role in metric fixed point theory. This paper explores some of the Banach space properties that can be associated with Property($M$) and Property($M^*$).
We prove that if a metric space $M$ has the finite CEP then $\mathcal F(M)\widehat{\otimes}_{\pi} X$ has the Daugavet property for every non-zero Banach space $X$. This applies, for instance, if $M$ is a Banach space whose dual is…
We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.
We investigate rich subspaces of $L_1$ and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products.
Some necessary and sufficient conditions are found for Banach function lattices to have the Radon-Nikod\'ym property. Consequently it is shown that an Orlicz space $L_\varphi$ over a non-atomic $\sigma$-finite measure space $(\Omega,…
In this paper we study set convergence aspects for Banach spaces of vector-valued measures with divergences (represented by measures or by functions) and applications. We consider a form of normal trace characterization to establish…